Smooth Maps Between Manifolds

Differential Geometry · Axiom Academy

LESSON Smooth Maps Between Manifolds Understanding how manifolds relate to each other through smooth mappings A map F: M → N between smooth manifolds is called smooth if for every point p ∈ M, there exist charts (U, φ) around p and (V, ψ) around F(p) such that the coordinate representation is smooth. The coordinate representation takes us from ℝⁿ to ℝᵐ via charts: ψ⁻¹ maps from ℝⁿ to the manifold M One of the most important properties of smooth maps is that compositions preserve smoothness . If F: M → N and G: N → P are smooth maps, then their composition G ∘ F: M → P is also smooth. This means smooth manifolds with smooth maps form a category , where: A diffeomorphism is a smooth map F: M → N that has a smooth inverse F⁻¹: N → M. Diffeomorphisms are the "isomorphisms" in the category of smooth manifolds. Properties of diffeomorphisms: Preserve smooth structure: M and N are "the same" as smooth manifolds Preserve dimension: If F: M → N is a diffeomorphism, then dim(M) = dim(N) Local property: Charts are examples of local diffeomorphisms Two special classes of smooth maps are defined by the rank of the differential (pushforward): Immersion: dim(M) ≤ dim(N), and rank(dF_p) = dim(M) Submersion: dim(M) ≥ dim(N), and rank(dF_p) = dim(N) Both: If F is both an immersion and submersion, it's a local diffeomorphism Let's examine some concrete examples that illustrate these concepts:

This is the written version of the interactive lesson above. See the full Differential Geometry course.